Fixing the dynamical evolution of self-interacting vector fields

Numerical simulations of the Cauchy problem for self-interacting massive vector fields often face instabilities and apparent pathologies. We explicitly demonstrate that these issues, previously reported in the literature, are actually due to the breakdown of the well posedness of the initial-value problem. This is akin to shortcomings observed in scalar-tensor theories when derivative self-interactions are included. Building on previous work done for $k$-essence, we characterize the well-posedness breakdowns, differentiating between Tricomi- and Keldysh-like behaviors. We show that these issues can be avoided by ``fixing the equations,'' enabling stable numerical evolutions in spherical symmetry. Additionally, we show that, for a class of vector self-interactions, no Tricomi-type breakdown takes place. Finally, we investigate initial configurations for the massive vector field which lead to gravitational collapse and the formation of black holes.